Geography Reference
In-Depth Information
7.4
INTERNAL GRAVITY (BUOYANCY) WAVES
We now consider the nature of gravity wave propagation in the atmosphere. Atmo-
spheric gravity waves can only exist when the atmosphere is stably stratified so that
a fluid parcel displaced vertically will undergo buoyancy oscillations (see Section
2.7.3). Because the buoyancy force is the restoring force responsible for gravity
waves, the term
buoyancy wave
is actually more appropriate as a name for these
waves. However, in this text we will generally use the traditional name
gravity
wave
.
In a fluid, such as the ocean, which is bounded both above and below, gravity
waves propagate primarily in the horizontal plane since vertically traveling waves
are reflected from the boundaries to form standing waves. However, in a fluid
that has no upper boundary, such as the atmosphere, gravity waves may propagate
vertically as well as horizontally. In vertically propagating waves the phase is a
function of height. Such waves are referred to as
internal
waves. Although internal
gravity waves are not generally of great importance for synoptic-scale weather
forecasting (and indeed are nonexistent in the filtered quasi-geostrophic models),
they can be important in mesoscale motions. For example, they are responsible for
the occurrence of mountain
lee waves
. They also are believed to be an important
mechanism for transporting energy and momentum into the middle atmosphere,
and are often associated with the formation of clear air turbulence (CAT).
7.4.1
Pure Internal Gravity Waves
For simplicity we neglect the Coriolis force and limit our discussion to two-
dimensional internal gravity waves propagating in the x, z plane. An expression
for the frequency of such waves can be obtained by modifying the parcel theory
developed in Section 2.7.3.
Internal gravity waves are transverse waves in which the parcel oscillations are
parallel to the phase lines as indicated in Fig. 7.8. A parcel displaced a distance δs
along a line tilted at an angle α to the vertical as shown in Fig. 7.8 will undergo
a vertical displacement δz
=
δs cos α . For such a parcel the
vertical
buoyancy
N
2
δz, as was shown in (2.52). Thus, the component
of the buoyancy force parallel to the tilted path along which the parcel oscillates
is just
force per unit mass is just
−
N
2
δz cos α
N
2
(δs cos α) cos α
(N cos α)
2
δs
−
=−
=−
The momentum equation for the parcel oscillation is then
d
2
(δs)
dt
2
(N cos α)
2
δs
=−
(7.24)
which has the general solution δs
i (N cos α) t ] . Thus, the parcels exe-
cute a simple harmonic oscillation at the frequency ν
=
exp [
±
=
N cos α. This frequency
